Title: Categorically generalizing bundles
Abstract: Vector bundles and principal bundles are important objects in differential geometry. Vector bundles over a manifold M are manifolds that are locally isomorphic to the Cartesian product M times V for some vector space V, while principal bundles over M are manifolds that are locally isomorphic to M times G for some Lie group G. I will explain some of the many ways to formulate these ideas categorically which generalize different aspects.
Tangent categories are the categorical generalization of the smooth structure giving rise to the tangent bundle. In them differential bundles generalize vector bundles.
On the other hand, join-restriction categories generalize the concept of locality from differential geometry and can be used to define principal bundles.
In the end I will give an outlook about how to generalize these concepts to infinity categories.